**Astroparticle Physics**

Volume 27, Issue 1, February 2007, Pages 92-98

### Abstract

A tachyonic spectral fit to the pulsed γ-ray flux of the Crab is performed, based on the superluminal spectral densities generated by ultra-relativistic electrons. The γ-ray wideband of the pulsar can be reproduced by a tachyonic cascade spectrum, capable of generating the spectral curvature in double-logarithmic plots as well as the extended spectral plateau defined by COMPTEL and EGRET flux points in the MeV band. Estimates on the magnetospheric electron population generating the radiation are inferred from the spectral fit, such as power-law indices, electron temperature and the electronic source count.

**Keywords: **Superluminal cascade
spectra; Tachyonic γ-rays; Longitudinal polarization; PSR B0531+21

**PACS classification codes: **95.30.Gv;
11.10.Lm; 97.60.Gb; 98.70.Sa

### Article Outline

### 1. Introduction

The goal is to point out evidence for the γ-ray spectrum of the Crab pulsar to be tachyonic. The most pronounced difference between electromagnetic and tachyonic radiation lies in the emission process. Due to the negative mass-square of the radiation field, there is a residual radiation that persists in the limit of large orbital bending radius, so that freely moving electrons can radiate superluminal quanta. Radiation densities attached to uniformly moving charges have no analog in electrodynamics, their very existence has substantial implications for the spacetime structure, requiring an absorber medium [1], [2] and [3]. The Green function of the superluminal radiation process is time-symmetric, there is no retarded propagator supported outside the lightcone. The advanced modes of the radiation field are converted into retarded ones by virtue of a nonlocal, instantaneous interaction with the absorber. The latter defines a universal frame of reference, necessary to render the superluminal signal transfer causal. The time symmetry of the Green function implies that there is no radiation damping, the radiating charge stays in uniform motion, the energy radiated is supplied by the oscillators constituting the absorber medium. The superluminal spectral densities depend on the velocity of the charge in the rest frame of the absorber. I also mention that tachyons and photons couple only indirectly via matter fields, there is no absorption of high-energy tachyons by the infrared background. At γ-ray energies, the speed of tachyons is very close to the speed of light, the noticeable difference to electromagnetic radiation is the longitudinally polarized flux component. The polarization of tachyons can be determined from transversal and longitudinal ionization cross-sections, which peak at different scattering angles.

These features of tachyonic wave propagation have been elaborated in previous papers, here we focus on a specific example, a tachyonic spectral fit to the pulsed γ-ray flux of the Crab. When tachyonic spectral densities are averaged with thermal or power-law electron densities, this leads to cascade spectra, extended spectral plateaus followed by power-law decay terminating in exponential decay. A glance at the γ-ray broadband of the Crab pulsar suggests that it can readily be fitted by a cascade spectrum, cf. the figures in Section 3.

In Section 2, we explain the transversal and longitudinal spectral densities of uniformly moving charges, as well as their averaging with thermal and exponentially cut power-law electron distributions. The spectral averaging can be done in closed form, and high- and low-temperature expansions of the tachyonic spectral densities are derived. The spectral breaks and slopes of tachyonic cascade spectra are discussed. In Section 3, we perform the spectral fit to the pulsed Crab γ-wideband, infer the cutoff temperature of the electronic/protonic source populations, and compare it to the break energies in the cosmic-ray spectrum. In Section 4, we present our conclusions.

### 2. Superluminal spectral densities and spectral averages

The transversal and longitudinal tachyonic radiation densities
of a uniformly moving electron read [3]

*Δ*

^{T}= 1 and

*Δ*

^{L}= 0.

*γ*is the electronic Lorentz factor,

*α*

_{q}the tachyonic fine structure constant, and

*m*

_{t}the tachyon mass. The spectral cutoff occurs at . Only frequencies in the range 0 <

*ω*<

*ω*

_{max}(

*γ*) can be radiated by a uniformly moving charge, the tachyonic spectral densities

*p*

^{T,L}(

*ω*) are cut off at the break frequency

*ω*

_{max}. The units =

*c*= 1 can easily be restored. We use the Heaviside–Lorentz system, so that

*α*

_{q}=

*q*

^{2}/(4

*πc*) ≈ 1.0 × 10

^{−13}. The tachyon mass is

*m*

_{t}≈ 2.15 keV/

*c*

^{2}, the tachyon–electron mass ratio

*m*

_{t}/

*m*≈ 1/238. These estimates are obtained from hydrogenic Lamb shifts [4]. The tachyonic count rates (tachyons emitted per unit time and unit frequency interval) are

*n*

^{T,L}(

*ω*)

*p*

^{T,L}(

*ω*)/

*ω*.

The radiation densities (1) refer to a
single charge with Lorentz factor *γ*. We average
them with electron distributions, power-laws exponentially cut with
Boltzmann densities, d*ρ* ∝ *E*^{−2−α}e^{−E/(kT)}d^{3}*p*,
or

*β*

*mc*

^{2}/(

*kT*). At

*α*= −2, the distribution is thermal. The electronic Lorentz factors range in an interval

*γ*

_{1}

*γ*< ∞. The normalization factors

*A*

_{α,β}(

*γ*

_{1},

*n*

_{1}) of densities (2) are determined by , where

*n*

_{1}is the electron count and

*γ*

_{1}the smallest Lorentz factor of the source population.

The averaging is carried out via

*ω*

_{max}(

*γ*) defined above. These averages can be reduced to the spectral functions

where and

*Δ*

^{T,L}as in (1).

The spectral range of the radiation densities (1) is
0 < *ω* < *ω*_{max}(*γ*).
Inversely, the condition
or
defines the minimal electronic Lorentz factor for radiation at this
frequency. (
stands for the rescaled frequency *ω*/*m*_{t}.)
That is, an electron in uniform motion can radiate at *ω*
only if its Lorentz factor exceeds .
The lower edge of Lorentz factors in the electronic source distribution
defines the break frequency, *ω*_{1} *ω*_{max}(*γ*_{1}),
or ,
which separates the spectrum into a low- and high-frequency band. We
have ,
and
if *ω* > *ω*_{1}.
The averaged energy density (3) can thus be
assembled as

*p*

^{T+L}

_{α, β}for the unpolarized density.

To obtain the low-temperature expansion, *β* 1, of the spectral average (5), we
substitute the asymptotic expansion of the incomplete *Γ*-function
[5] into the
spectral function *B*^{T,L}(*ω*;*γ*;*α*,*β*),

*βγ*1. The high-temperature expansion,

*β*1, of

*B*

^{T,L}in (4) is based on the ascending series of

*Γ*(−

*α*− 1,

*βγ*),

which requires

*βγ*1. In the opposite limit,

*βγ*1, we may still use the low-temperature expansion (6), even if

*β*1. Singularities occurring at integer

*α*in (7) cancel if

*ε*-expanded.

If *γ*_{1} = 1,
*β* 1
and *α* > 1, the maximum of *p*^{T,L}(*ω*;
*γ*_{1}, *n*_{1})_{α, β},
cf. (5), is
determined by the lead factor
in the high-temperature expansion (7). This peak
is located at
and is followed by power-law decay ∝*ω*^{−α}
for *ω* *m*_{t}/*β*
and exponential decay starting at about *ω* ≈
*m*_{t}/*β*. The
low-temperature expansion (6) applies for *ω* *m*_{t}/*β*.
If *γ*_{1} 1 (but still *βγ*_{1} 1
and *α* > 1), the peak of *p*^{T,L}(*ω*;*γ*_{1}, *n*_{1})_{α, β}
is determined by the factor
in (7) and occurs
at *ω*_{max} ≈ *m*_{t}.
It is followed by a broken power-law, at first *p*^{T,L}_{α, β} ∝ 1/*ω*,
in the range 1 *ω*/*m*_{t} *γ*_{1},
and then *p*^{T,L}_{α, β} ∝ *ω*^{−α}
in the interval *γ*_{1} *ω*/*m*_{t} 1/*β*.
At *ω* ≈ *m*_{t}/*β*,
there is an exponential cutoff according to (6).

The spectral fit is performed with the *E*^{2}-rescaled
flux density *E*^{2}d*N*^{T,L}/d*E*,
which relates to the energy density *p*^{T,L}_{α, β}
in (5) as

*d*is the distance to the pulsar. The preceding scaling relations for

*p*

^{T,L}

_{α, β}give a plateau value

*E*

^{2}d

*N*

^{T,L}/d

*E*∝ 1 in the range 1

*E*/(

*m*

_{t}

*c*

^{2})

*γ*

_{1}, followed by power-law decay

*E*

^{2}d

*N*

^{T,L}/d

*E*∝

*E*

^{1−α}in the band

*γ*

_{1}

*E*/(

*m*

_{t}

*c*

^{2}) 1/

*β*, and subsequent exponential decay.

In the foregoing discussion of the high-temperature
asymptotics, we assumed an electron index *α* > 1.
We still have to settle the case *α* < 1.
The spectral peak at *ω*_{max} ≈ *m*_{t}
is again determined by the factor .
Adjacent is a power-law slope *p*^{T,L}_{α,β} ∝ 1/*ω*
in the range 1 *ω*/*m*_{t}
1/*β*, so that *E*^{2}d*N*^{T,L}/d*E* ∝ 1.
At *ω* ≈ *m*_{t}/*β*,
there is the cross-over to exponential decay, cf. (6). This holds
true for *γ*_{1} = 1
as well as *γ*_{1} 1, provided *βγ*_{1} 1,
since the leading order of the high-temperature expansion (7) does not
depend on *γ* for *α* < 1.
If *βγ*_{1} 1 despite of *β*
being small, then the low-temperature expansion (6) applies
instead of (7).

### 3. Tachyonic cascade spectra and spectral fits

Fig. 1, Fig. 2 and Fig. 3 show the *E*^{2}-scaled
differential number flux, *E*^{2}d*N*^{T+L}/d*E*,
as defined in (8), fitted to
the data points. As the polarization is not observed, we add the
transversal and longitudinal densities, writing *p*^{T+L} *p*^{T} + *p*^{L} for
the energy density and d*N*^{T+L} d*N*^{T} + d*N*^{L}
for the number flux. We restore the natural units on the right-hand
side of (8), up to now
we have used = *c* = 1,
and write *E*_{(1)} for *ω*_{(1)}.
The spectral maps are in MeV units, with the differential tachyon flux d*N*^{T,L}/d*E*
in units of MeV^{−1}s^{−1}cm^{−2}.
The normalization factor in (5) is
dimensionless, ,
where *n*_{1} is the number of
radiating electrons with Lorentz factors exceeding *γ*_{1},
distributed according to density (2). *α*
is the electronic power-law index, and *β* the
exponential cutoff in the electron spectrum, both to be determined from
the spectral fit like *n*_{1} and
*γ*_{1}. As for the electron count
*n*_{1}, it is convenient to use a
rescaled parameter
for the fit,

*m*

_{t}

*c*

^{2}≈ 2.15 in the spectral function (4). The purpose of this rescaling is to avoid the distance estimate in the actual fitting procedure. Once is extracted from the spectral fit, we can calculate via (9) the number of radiating electrons

*n*

_{1}constituting the density d

*ρ*

_{α,β}. This, however, implies that all tachyons reaching the detector are properly counted. At γ-ray energies, only a tiny

*α*

_{q}/

*α*

_{e}-fraction (the ratio of tachyonic and electric fine structure constant,

*α*

_{q}/

*α*

_{e}≈ 1.4 × 10

^{−11}) of the tachyon flux is actually absorbed, cf. [6] and Section 4. This requires a rescaling of the electron count

*n*

_{1}, so that the actual number of radiating electrons is . In Table 1, we list the flux amplitude inferred from the spectral fit, as well as the renormalized electron count . (In the table, we drop the subscript 1 in and .) The density d

*ρ*

_{α,β}is defined by parameters obtained from the spectral fit.

Fig. 1. Tachyonic spectral map of the Crab pulsar.
PDS, COMPTEL and EGRET (open squares) data points from [7], GRIS points
from [8], OSSE points
from [9]; also see [10], [11] and [12] for COMPTEL
data on the pulsed Crab emission. The open squares are the final EGRET
data covering cycles 1–4, corrected for spark chamber gas aging [13]. For
comparison, we also indicate the EGRET flux points based on cycles 1–3,
cf. [14]; the error
bars on the last four flux points are suppressed, they are similar to
those on the open squares with the same abscissa. In the *χ*^{2}-fit,
only the final EGRET data (open squares) are considered, and the
vertical error bar on the second flux point is assumed to hold for the
subsequent six points as well. The solid line T + L
is the unpolarized differential tachyon flux d*N*^{T+L}/d*E*,
rescaled with *E*^{2} for better
visibility, cf. (12). The total
flux T + L can be split into transversal (T,
dot–dashed) and longitudinal (L, dot–dot–dashed) flux components d*N*^{T,L}/d*E*,
cf. (13). The
cascade spectra labeled *ρ*_{1,2}
(dashed) are the unpolarized flux densities d*N*^{T+L}(*E*,*ρ*_{i})/d*E*
of two electron populations *ρ*_{i=1,2},
cf. Table 1 and (14). These
cascades add up to the total flux T + L, which is the
actual spectral fit with parameters (of the electron densities *ρ*_{i=1,2})
listed in Table 1. The MeV
plateau terminates in a nearly straight GeV power-law slope. In
contrast to the *ρ*_{2}-cascade,
the cutoff temperature of *ρ*_{1}
is too high to significantly curve the power-law slope in the energy
range covered here, cf. Fig. 3 for the
wideband view. A spectral break at *E* ≈ *m*_{t}*γ*_{1} ≈ 1.3 GeV
is visible as edge in the longitudinal spectral map, cf. after (8). *ρ*_{1}
is the cascade producing the MeV plateau and the adjacent GeV power-law
slope ∝1/*E* (*α* = 1,
cf. Table 1). The
soft *γ***-**ray band is
generated by adding the flux of the *ρ*_{2}-cascade
to the *ρ*_{1}-plateau. The
spectral curvature is caused by the Boltzmann factor in the averaged
spectral densities, whose cutoff frequency is *E*_{cut} = (*m*_{t}/*m*)*kT*,
defined by
according to the low-temperature/high-frequency expansion (6). This
implies cutoffs at 88 GeV for the *ρ*_{1}-cascade
and 6.3 MeV for *ρ*_{2}.
The cutoff temperature of the respective electron densities is recorded
in Table 1. This
figure is a close-up of the γ-wideband in Fig. 3.

Fig. 2. Residuals of the least-squares fit in Fig. 1. The
cascade *ρ*_{1} in Fig. 1 is based
on ten EGRET flux points; the ninth point above the 10^{−4}-mark
is discounted in the *χ*^{2}-fit.
This cascade is fitted with *χ*^{2} ≈ 3.85
(unreduced) and a confidence level of 57% with 5 dof. The cascade
depends nonlinearly on the four fitting parameters, which limits the
significance of Gaussian *p*-values, *Γ*(dof/2,*χ*^{2}/2)/*Γ*(
dof/2), apart from the low statistics. For instance, the first seven
points of the EGRET plateau amount to a horizontal straight-line fit
effectively depending on only one independent fitting parameter, which
suggests a much higher confidence level. The *ρ*_{2}-cascade
in Fig. 1 on top of
*ρ*_{1} is based on the COMPTEL
points. The first and last of the indicated 12 COMPTEL points are
discarded in the fit, and *χ*^{2}
is weighted with the indicated error bars. We find *χ*^{2} ≈ 1.40
and a confidence level of 97% with 6 dof. This high *p*-value
is mainly due to the larger error bars, as compared to the EGRET fit.
The PDS, GRIS and OSSE points are not included in the *χ*^{2}-fit
of the *ρ*_{2}-cascade, but it is
evident from the rather symmetrically spaced residuals that the cascade
is a good fit to these dispersed data sets. The residuals (data points
minus fitting function (12), (13), (14) and (15) with
parameters as in Table 1) have
not been normalized.

Fig. 3. γ-Ray wideband of the Crab pulsar. Data
points depicted as open symbols are referenced in Fig. 1. Filled
symbols (upper bounds) from CELESTE [15], STACEE [16], Whipple [17], CAT [18], and HEGRA [19]. The
spectral fit T + L (solid line) is done with the
rescaled unpolarized tachyon flux *E*^{2}d*N*^{T+L}/d*E*
as in Fig. 1. The GeV
power-law slope following the spectral break at 1.3 GeV terminates in
exponential decay. The GeV-fit hinges on the last EGRET point, the
CELESTE and STACEE points, and the first CAT point. The latter three
data points are upper limits. Another option for the high-energy fit is
the dotted extension (T + L)_{B}
of the MeV plateau, fitted to the EGRET points only, the upper bounds
are evidently satisfied. This fit is based on a thermal Boltzmann
distribution generating the *ρ*_{1}-cascade,
the parameters of the electron density *ρ*_{1}
in Table 1 are
replaced by *α* = −2, *β* ≈ 7.17 × 10^{−7}
and *γ* = 1, and
remains unchanged. In this case, there is no power-law interface
between the MeV plateau and exponential decay. Presently, there are not
enough data points in the GeV band to discriminate between the
exponentially cut power-law fit T + L and the thermal
fit (T + L)_{B}.

Electronic source densities *ρ*_{i}
generating the tachyonic cascade spectra of the Crab pulsar

Each distribution is defined by parameters ,
cf. Eq. (2). *α*
is the electronic power-law index, *β* the cutoff
parameter in the Boltzmann factor, and *γ* the lower
edge of Lorentz factors in the source population.
is the amplitude of the tachyonic flux density generated by *ρ*_{i}.
The electron count *n*^{e} is
inferred from
via (9) and *d* ≈ 2 kpc.
*kT* is the electron temperature. The *ρ*_{1,2}-cascades
in Fig. 1 and Fig. 3 are
obtained by averaging the superluminal radiation densities (1) with the
electron distributions *ρ*_{1,2}
listed here. The parameters
are extracted from the spectral fit.

In the case of a thermal electron distribution (*α* = −2),
we may identify *kT*
[TeV] ≈ 5.11 × 10^{−7}/*β*.
We use this definition of electron temperature also for other electron
indices *α*. *kT* is the energy at
which the exponential decay of the electron density (2) sets in. At
high temperature, density (2) is peaked at
*γ*_{peak} ≈ −*α*/*β*,
provided *α* < 0 and *β* *α*; the peak is
followed by exponential decay. If *α* > 0,
the electron density is decreasing monotonically, at first as power-law
and exponentially beyond *kT*. Nearly straight
power-law slopes, as seen in the low GeV range of Fig. 1, can only
emerge for electron indices *α* > 1,
cf. after (7). The
spectral break from the MeV plateau into the power-law slope is
determined by the lower edge *γ*_{1}
of electronic Lorentz factors, cf. after (4). If *α* < 1,
the MeV plateau is followed by exponential decay, without power-law
cross-over, as exemplified by the thermal cascade
(T + L)_{B} in Fig. 3.

Wideband cascade spectra are obtained by averaging over
multiple electron populations,

*A*

_{αi,βi}(

*γ*

_{i},

*n*

_{i}) of the individual averages in this series are calculated via

where

*n*

_{i}is the electron count and

*γ*

_{i}the smallest Lorentz factor in the population defined by d

*ρ*

_{αi,βi}. The electron numbers

*n*

_{i}(attached to a power-law index

*α*

_{i}, temperature

*β*

_{i}=

*mc*

^{2}/(

*kT*

_{i}), and interval boundary

*γ*

_{i}) determine the weights in the linear combination (10).

We summarize the flux density plotted in Fig. 1 and Fig. 3. The
spectral fit is done for unpolarized radiation, so that we start by
summing over the polarization components,

*E*

^{2}-scaled flux density

*E*

^{2}d

*N*/d

*E*, as well as the polarization components

*E*

^{2}d

*N*

^{T,L}/d

*E*. (We occasionally use a superscript T + L to indicate the total flux density, writing d

*N*

^{T+L}instead of d

*N*.) The polarization components in (12) are obtained by summing over two cascades, cf. (8) and (10),

The individual cascades used in the least-squares fit described in the caption of Fig. 2 are assembled from (2), (4), (5) and (8). Restoring dimensions, we find their explicit shape as

Each cascade depends on four fitting parameters,

*α*

_{i},

*β*

_{i},

*γ*

_{i},

*n*

_{i}, whose meaning is explained at the beginning of this section. [The actual spectral fit described in the figure captions is done by at first summing over the polarizations in (14) and by adding the

*χ*

^{2}-fits of the unpolarized cascades

*E*

^{2}d

*N*

^{T+L}(

*E*;

*α*

_{i},

*β*

_{i},

*γ*

_{i},

*n*

_{i})/d

*E*. That is, we interchange the summations in (12) and (13) to obtain

*E*

^{2}d

*N*/d

*E*.]

*θ*is the Heaviside step function, and we use the shortcut

where we put

*Δ*

^{T}= 1 and

*Δ*

^{L}= 0, for transversal and longitudinal polarization, respectively.

The total tachyonic flux density *E*^{2}d*N*/d*E*
is plotted in Fig. 1 and Fig. 3 (solid
line T + L). The individual unpolarized cascades, *E*^{2}d*N*^{T+L}(*E*;*α*_{i},
*β*_{i}, *γ*_{i},
*n*_{i})/d*E*,
obtained by summing over the polarizations in (14), are
labeled *ρ*_{i=1,2}
(dashed curves). They add up to the total tachyonic flux density
labeled T + L. The electron densities *ρ*_{i}
in Table 1 produce
the cascade spectra *ρ*_{i}
and are in turn defined by parameters
obtained from the least-squares fit.

The cutoff *kT* in the electron energy is
listed in Table 1, for
each electron population generating a cascade in the γ-ray wideband. In
the case of protonic source particles, the cutoff energies have to be
multiplied with 1.8 × 10^{3}, the
proton/electron mass ratio. These cutoffs are to be compared to two
spectral breaks in the cosmic-ray spectrum, occurring at 10^{3.5}
TeV and 10^{5.8} TeV, dubbed knee and second knee,
respectively [23]. A protonic
source density generating the *ρ*_{1}-cascade
has a cutoff at about 10^{4.6} TeV, suggesting
protons in the pulsar magnetosphere with energies between the first and
second knee.

### 4. Conclusion

We have considered superluminal spectral densities generated by freely moving electrons, discussed the spectral averaging with electronic source distributions, and derived the high- and low-temperature expansions of the averaged radiation densities. We demonstrated that the γ-ray flux of the Crab pulsar can be reproduced by tachyonic cascade spectra, and explained the spectral plateau, the power-law slopes, and the spectral curvature.

Here, we have studied superluminal radiation from electrons in uniform motion. Tachyonic synchrotron and cyclotron radiation was investigated in [24] and [25]. In the zero magnetic field limit, the tachyonic synchrotron densities converge to the spectral densities (1) for uniform motion. Electromagnetic synchrotron radiation vanishes in this limit, of course. A finite bending radius induces modulations in the spectral densities. These oscillations are quite small, just tiny ripples along the spectral slope with increasing amplitude toward the spectral break [25]. If integrated over a thermal or power-law electron distribution as done here, these oscillations are averaged out.

Several electromagnetic radiation mechanisms have been invoked to model the spectra of γ-ray pulsars, most notably curvature radiation due to electric fields followed by pair creation, accompanied by synchrotron radiation and inverse Compton scattering [26]. These mechanisms are quite interlocked, cf. the flow-chart in [27], and subject to frequent revisions [28] and [29]. An electromagnetic spectral fit to the pulsed Crab emission, directly comparable to Fig. 1, can be found in Fig. 11 of [28].

In Section 3, we pointed
out that only a fraction *α*_{q}/*α*_{e} ≈ 1.4 × 10^{−11}
of the tachyon flux arriving at the detector is actually absorbed, so
that the measured flux has to be rescaled with the ratio of electric
and tachyonic fine structure constant. This rescaling applies to γ-ray
spectra only, to frequencies much higher than the 2 keV
tachyon mass, so that the mass-square can be dropped in the tachyonic
dispersion relation. At X-ray energies, we have to rescale with the
respective cross-section ratio, *σ*_{e}(*ω*)/*σ*_{q}(*ω*),
where *σ*_{e} is the
electromagnetic cross-section and *σ*_{q}
its tachyonic counterpart. The detection mechanism for hard and
high-energy X-rays is usually ionization, generating scintillations in
NaI(Tl) and CsI(Na) crystals of phoswich detectors [30], or gas
scintillations by ionization of Xenon atoms in proportional counters [31]. Tachyonic
ionization cross-sections of hydrogen-like ground states, that can be
used for X- as well as γ-rays, have been calculated in [32]. At γ-ray
energies, photonic and tachyonic dispersion relation coincide, and the
cross-section ratio of the detection mechanism (such as ionization [33], Compton
scattering [34] and pair
production [35], the latter
two quantified by Klein-Nishina and Bethe–Heitler cross-section)
approaches a constant limit value *α*_{e}/*α*_{q},
which amounts to an overall rescaling of the tachyonic flux amplitude.
By contrast, at X-ray energies, we would have to rescale the tachyonic
flux density with the energy dependent cross-section ratio, which
affects the shape of the spectral map. If a grating spectrometer is
used, the interference peaks are determined by wavelength. In this
case, the observational plots have to be reparametrized by wavelength,
and then the tachyonic dispersion relation is substituted to obtain the
actual energy-flux relation.

### Acknowledgement

The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.